Dynamically tilting flat table to impart a time-varying gravity-induced acceleration on a floating spacecraft simulator

ABSTRACT

Disclosed is a planar test bed comprising a planar surface and further comprising mechanical couplings in mechanical communication with the planar table and the supporting legs. The mechanical couplings are translatable to provide three degrees of freedom for orientation of the planar surface. A processor receives position and velocity information describing an object on the planar surface, and calculates a relative acceleration typically using a function a R =f(t,x R ,v R ,μ t ). The processor communicates with the mechanical couplings to establish an orientation where a local gravity vector projects onto the planar surface and generates acceleration with magnitude and direction substantially equal to the desired acceleration a R  The operations occur in cyclic fashion so the desired accelerations and planar orientations are updated as an object transits over the planar surface.

RELATION TO OTHER APPLICATIONS

This patent application claims the benefit of U.S. ProvisionalApplication No. 62/445,775 filed Feb. 7, 2017, which is herebyincorporated in its entirety.

FIELD OF THE INVENTION

One or more embodiments relates generally to a planar test bed with aprogrammed controller specifying time-variant orientations of a planarsurface on the test bed via communication with adjustable mounts. Theplanar surface of the test bed may be utilized for the simulation ofspacecraft using test vehicles supported on air-bearings and transitingon the planar surface.

BACKGROUND

Floating spacecraft simulators (also known as planar air bearing testbeds) are extensively utilized to conduct spacecraft dynamic model andguidance and control systems development, validation, and verificationin a dynamically representative environment. In a floating spacecraftsimulator, one or more robotic test vehicles, each of them representinga spacecraft, operate on top of a smooth, flat, and horizontally leveledsurface. Planar air bearings are used to create a low frictionloadbearing interface between the test vehicle and the flat surface.Given the smoothness and horizontally of the flat surface, the testvehicles experience a quasi-frictionless and weightless motion in twodimensions. This dynamic environment recreates the drag-free andweightless motion experienced by orbiting spacecraft (commonly referredto as reduced gravity or microgravity) on a plane.

The planar air bearings, which can be mounted either on the testvehicles or on the flat surface, use pressurized air to establish a thinfilm of air between themselves and their opposing surface. The air filmacts as a lubricant and supports the weight of the test vehicle. Thereduced gravity level attained on the test bed depends on the airbearings performance as well as on the flatness, smoothness, andhorizontality of the operating surface. Epoxy-coated floors, glasspanes, and granite tables are commonly used as operating surfaces.Granite tables are usually the preferred option as they can beaccurately leveled and machined to a high planarity and smoothnesslevel, while offering excellent stiffness and thermal stability.

Surveys on the technology and applications of air-bearing test beds areavailable. See e.g., Schwartz et al., “Historical review of air-bearingspacecraft simulators,” Journal of Guidance, Control, and Dynamics 26(4) (2003); see also Rybus et al., “Planar air-bearing microgravitysimulators: Review of applications, existing solutions and designparameters,” Acta Astronautica 120 (2016). See also, Wilde et al.,“Experimental Characterization of Inverse Dynamics Guidance and Controlin Docking with a Rotating Target,” Journal of Guidance, Control, andDynamics 39(6) (2016); see also Ciarcia et al., “Near-optimal guidancefor cooperative docking maneuvers,” Acta Astronautica 102 (2014); seealso Curti et al., “Lyapunov-based Thrusters' Selection for SpacecraftRotational and Translational Control: Analysis: Simulations andExperiments,” Journal of Guidance, Control, and Dynamics 33(4) (2010);see also Romano et al., “Laboratory Experimentation of AutonomousSpacecraft Approach and Docking to a Collaborative Target,” Journal ofSpacecraft and Rockets 44(1) (2007). In addition to air-bearingfacilities, neutral buoyancy, free-falling (either with parabolicflights or drop towers), and suspension apparatus for gravitycompensation are also used in an attempt to recreate a reduced gravityenvironment. See e.g. Menon et al., “Issues and solutions for testingfree-flying robots,” Acta Astronautica 60 (12) (2007); and see Xu etal., “Survey of modeling, planning, and ground verification of spacerobotic systems,” Acta Astronautica 68 (11-12) (2011).

Having ground-based facilities where the dynamics of orbiting spacecraftcan be recreated is exceptionally valuable to develop, test, andvalidate spacecraft dynamic model and guidance and control systems. Inparticular, these test-beds are mainly used to recreate spacecraftdocking and short duration close proximity maneuvers. Other potentialapplications, that would greatly expand the applicability of the testbed, include, for example, recreating spacecraft rendezvous, longerduration proximity maneuvers, and planetary landings. In these otherapplications, the dynamic environment to be recreated includesnon-inertial, gravitational or other environmental accelerations thatcannot be neglected. In principle, the test vehicle's actuators could beused to recreate these accelerations, but using the test vehicle'sactuators interferes with the dynamic behavior of the test vehicle,reducing the dynamic fidelity of the test bed. See e.g. Ciarcia et al.,“Emulating Scaled Clohessy-Wiltshire Dynamics on an Air-bearingSpacecraft Simulation Testbed,” AIAA SciTech Forum, Guidance,Navigation, and Control Conference, 9-13 Jan. 2017, Grapevine, Tex.

By tilting the test bed operating surface away from its nominalhorizontally leveled state a gravitational acceleration is imparted tothe test vehicles. This gravitational acceleration, if carefullycontrolled, can be used to recreate non-inertial, gravitational or otherenvironmental accelerations. For example, the recreation of the surfacegravity of a planetary body (e.g., asteroid) may be obtained toexperimentally evaluate spacecraft landing maneuvers. By staticallytilting the operating surface, a constant acceleration in a particulardirection will be imparted to the test vehicle, thus recreating theplanetary body's surface gravity. When recreating the relative motion ofthe two orbiting spacecraft in close proximity the resultingnon-inertial acceleration is time varying and depends on the relativestate between the two vehicles, thus requiring a time-varying operatingsurface tilt. Other environmental accelerations (e.g., solar radiationpressure, aerodynamic drag, gravitational acceleration while orbiting anirregular central body) are also be time-varying and require atime-varying operating surface tilt.

In current air bearing test bed setups, the orientation of the operatingsurface with respect to the local horizontal may, in some instances, bemanually adjustable. Tilting the operating surface is then a manual,discrete operation, and the acceleration provided by gravity acts withconstant direction and magnitude regardless of changing spatial andkinematic conditions among the test vehicle or vehicles present.

It would be advantageous to provide a floating spacecraft simulatorcomprising a flat table that dynamically and automatically changes itsorientation with respect to the local gravity to impart a desiredtime-varying acceleration to the test vehicles. This capability could beused to impart time-varying accelerations to the test vehicles withoutusing their actuators, thus allowing to recreate, for example,rendezvous, proximity maneuvers, and planetary landings, greatlyextending the applicability of planar air bearing test beds.

These and other objects, aspects, and advantages of the presentdisclosure will become better understood with reference to theaccompanying description and claims.

SUMMARY

The planar test bed disclosed comprises a planar table comprising aplanar surface and further comprising one or more mechanical couplings,with each mechanical coupling in mechanical communication with theplanar table and a supporting leg. The mechanical couplings areslidably, rotatably, or otherwise translatable to provide three degreesof freedom to the planar table. A processor is in data communicationwith the mechanical couplings and issues commands to the mechanicalcouplings to alter the spatial orientation of the planar table in atransient process, in order to allow the projection of the local gravityvector g onto the operating surface plane to mimic an anticipatedrelative acceleration imparted to a test vehicle in a simulatedenvironment.

The processor is programmed to provide commands to the mechanicalcouplings by executing a series of programmed steps in a generallycyclic fashion. At a given time, the processor receives, typically amongother relevant parameters, an object location describing the location ofan object on the planar surface, and also receives an object velocity.Using the position and velocity, the processor calculates a relativeacceleration desired between the object and a fixed location on theplanar table typically using a function a_(R)=f(t,x_(R),v_(R),μ_(t)),where a_(R) denotes the relative acceleration vector, t the time, x_(R)the relative position vector, v_(R) the relative velocity vector andμ_(t) a set of time-varying parameters. Given the desired acceleration,the processor issues commands to the mechanical couplings andestablishes the planar surface in an orientation whereby a vectorprojection of a local gravity vector onto the planar surface generatesan acceleration parallel to the planar surface and having magnitude anddirection substantially equal to the desired acceleration a_(R). Theplanar test bed conducts this process in a cyclic manner so that desiredaccelerations and planar orientations are updated as an object transitsover the planar surface. In a particular embodiment, the system furthercomprises a camera vision sensing system providing at least positionaldata to the processor during the cyclic process.

The novel apparatus and principles of operation are further discussed inthe following description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a particular embodiment of a planar test bed.

FIG. 2 illustrates a coordinate system and scenario for simulation onthe planar test bed.

FIG. 3 illustrates an exemplary scenario on a planar surface.

FIG. 4 illustrates a particular orientation of a planar surface.

FIG. 5 illustrates an embodiment of a method performed by a planar testbed system.

FIG. 6 illustrates a specific embodiment of a planar test bed.

FIG. 7 illustrates another embodiment of a planar test bed.

FIG. 8 illustrates an additional embodiment of a planar test bed.

Embodiments in accordance with the invention are further describedherein with reference to the drawings.

DETAILED DESCRIPTION OF THE INVENTION

The following description is provided to enable any person skilled inthe art to use the invention and sets forth the best mode contemplatedby the inventor for carrying out the invention. Various modifications,however, will remain readily apparent to those skilled in the art, sincethe principles of the present invention are defined herein specificallyto provide an apparatus and method for the generation of a time-varyinggravity-induced acceleration on a test vehicle operating on top of theplanar test bed surface.

A particular embodiment of the planar test bed is illustrated at FIG. 1and indicated generally at 100. Planar test bed 100 comprises planartable 101, with planar table 101 comprising planar surface P. Planarsurface P further comprises first location 102, where first location 102is a fixed location within the boundary of planar surface P, and furthercomprises an axis x_(p) and an axis y_(p), with the axis x_(p)perpendicular to the axis y_(p). Planar test bed 100 additionallycomprises at least one mechanical coupling such 103, 105, and 107 inmechanically communication with planar table 101 and supporting legs104, 106, and 108 respectively. In operation, supporting legs 104, 106,and 108 are typically standing on a fixed floor. A processor 109 is indata communication with the mechanical couplings and issues commands tothe mechanical couplings to alter the spatial orientation of planartable 101 in a transient process, in order to allow the projection ofthe local gravity vector g onto the operating surface plane to mimic ananticipated relative acceleration on the test vehicle in a simulatedenvironment. For reference. FIG. 1 also illustrates a point on planarsurface P comprising an origin O, with an axis z₀ parallel to a localgravity vector g, an axis x₀ perpendicular to the axis z₀, and an axisy₀ perpendicular to both x₀ and z₀.

As discussed, test beds involving planar surfaces and intended tosupport test vehicles on air bearings have been used to simulatefree-flying spacecraft environments. The air bearings allowsquasi-frictionless translational and rotational freedom of the testvehicle over the planar surface, and often provide for supporting legadjustment in order to level the planar surface with the localhorizontal, achieving a reduced gravity environment on the plane. Seee.g., Rybus, et al., “Planar air-bearing microgravity simulators: Reviewof applications, existing solutions and design parameters,” ActaAstronautica 120 (2016), among others. However, these planar test bedsare established in a given, fixed orientation so that the simulatedacceleration on the planar surface is some constant value independent oftest vehicle location. This presents significant limitation in systemswhere a relative acceleration imparted to the test vehicle is expectedto vary based, for example, on the test vehicle position and velocity.For example, FIG. 2 illustrates time-based positions of a targetspacecraft and a chaser spacecraft orbiting a body B and shown for timest₁, t₂, t₃, and t₄, with the position of the target spacecraft at thevarious times indicated as T(t₁), T(t₂), T(t₃), and T(t₄) respectively.The orbital path of the target spacecraft is indicated as K, and therespective radius vectors of the target from B at the respective timesare illustrated as r_(T)(t₁), r_(T)(t₂), r_(T)(t₃), and r_(T)(t₄).Additionally indicated are the positions of a chaser spacecraftexperiencing a different path around body B, with positions indicated asC(t₁), C(t₂), C(t₃), and C(t₄). A series of relative position vectorsdenoted ρ(t), ρ(t₂), ρ(t₃), and ρ(t₄) describe the location of thechaser spacecraft relative to the target spacecraft at the indicatedtimes. At FIG. 2, the relative position vectors can be resolved onto aCartesian Coordinate System which is local-vertical, local-horizontal(LVLH) and attached to the target spacecraft, where the coordinatesystem rotates with the target's radius vectors r_(T)(t₁), r_(T)(t₂),r_(T)(t₃), and r_(T)(t₄), and comprises an axis y_(R) in the radialdirection and an axis x_(R) along the direction of the target's orbitalvelocity. A third axis z_(R) (not shown), perpendicular to the orbitalplane, may also be described to complete a right-handed orthogonaltriad. The relative position vectors ρ(t₁), ρ(t₂), ρ(t₃), and ρ(t₄)describe the relative positions between the target and the chaser asstated, but additionally and as is understood, the time-varying natureof the relative position vectors provides relative velocity {dot over(ρ)}(t) and relative acceleration {umlaut over (ρ)}(t) between the twobodies.

The relative acceleration {umlaut over (ρ)}(t) between the two bodiescan be seen as the superposition of multiple accelerations caused bymultiple physical phenomena. First, there is the acceleration betweenthe two spacecraft arising from non-inertial nature of the referenceframe and from the relative orbital dynamics {umlaut over (ρ)}_(rod)(t).The relative dynamics acceleration {umlaut over (ρ)}_(rod)(t) can bemodelled, in the case of circular orbits and for small relative orbitaldistances, with the Hill-Clohessy-Wiltshire equations. In more generalterms, the relative orbital dynamics acceleration between the two bodiesmight be expressed by some function {umlaut over(ρ)}_(rod)(t)=f(t,ρ,ρ,μ_(e)), with μ_(rod) denoting the relevant orbitalparameters, such as: mean motion, eccentricity, true anomaly, etc. Inaddition to the acceleration caused by the relative dynamics, the targetor chaser spacecraft may be subject to a relative acceleration {umlautover (ρ)}_(e)(t) caused by other environmental disturbances, such as:aerodynamic drag, solar radiation pressure, etc. This relativeacceleration between the two bodies might also be expressed by somefunction {umlaut over (ρ)}_(e)(t)=f(t,ρ,{dot over (ρ)},μ_(e)), withμ_(e) denoting the relevant environmental parameters (e.g., atmosphericdensity, orbital elements, spacecraft attitude, etc.). Finally, theaction of the chaser or target actuators may also cause a relativeacceleration {umlaut over (ρ)}_(a)(t). The resulting relativeacceleration is then the combination of these {umlaut over(ρ)}(t)={umlaut over (ρ)}_(rod)(t)+{umlaut over (ρ)}_(e)(t)+{umlaut over(ρ)}_(a)(t). The relative acceleration {umlaut over (ρ)}_(a)(t) isrecreated by the chaser and target actuators, while the relativeacceleration due to the relative orbital dynamics {umlaut over(ρ)}_(rod)(t) or the environmental forces {umlaut over (ρ)}_(e)(t) isrecreated by tilting the test bed operating surface {umlaut over(ρ)}_(t)(t)={umlaut over (ρ)}_(rod) (t)+{umlaut over (ρ)}_(e)(t). Thisacceleration to be provided by the operating surface tilt might beexpressed by some function {umlaut over (ρ)}_(t)(t)=f(t,ρ,{dot over(ρ)},μ_(t)), with μ_(t) denoting the relevant orbital μ_(rod) andenvironmental parameters μ_(e).

As is additionally understood, the relative position vectors ρ(t),ρ(t₂), ρ(t₃), and ρ(t₄) between the target and the chaser can berepresented on a planar test bed surface such as that illustrated atFIG. 3, with planar test bed 301 comprising planar surface P and furthercomprising first location 302, where first location 302 is a location onplanar surface P. Here, the fixed location may be a static location onplanar surface P, or may vary transiently in a manner known to acontroller directing the orientation of the planar surface P. Thevarious relative position vectors ρ(t₁), ρ(t₂), ρ(t₃), and ρ(t₄)expected from FIG. 2 are represented on planar surface P and relative tofirst location 302 as ρ_(S)(t₁), ρ_(S)(t₂), ρ_(S)(t₃), and ρ_(S)(t₄),with x_(R) and y_(R) illustrated for reference. An absolute orthogonalcoordinate frame is additionally illustrated at FIG. 3 as x_(o), y_(o),and z_(o), where z_(o) is parallel to a local gravity vector andproceeding into the page. Adjusting the orientation of test bed 301relative to x_(o), y_(o), and z_(o) will provide a projection of thelocal gravity vector on planar surface P to simulate a given relativeacceleration expected between a first body represented by first location302 and a second body represented by positions described by ρ_(S)(t₁),ρ_(S)(t₂), ρ_(S)(t₃), and ρ_(S)(t₄). However, because the relativeacceleration to be simulated by planar surface P is some functionpt(t)=f(t,ρ,{dot over (ρ)},μ_(t)), without a substantially continuousmeans by which a necessary orientation may be determined and provided,the assumed orientation will have validity under only a single set ofconditions. For example, an orientation of planar surface P assumed forthe position described by ρ_(S)(t₁) may provide a simulated relativeacceleration appropriate for when a test vehicle assumes the ρ_(S)(t₁)position and relative velocity, however the orientation becomessignificantly less accurate as the test vehicle attempts to emulate theconditions at ρ_(S)(t₂), ρ_(S)(t₃), and ρ_(S)(t₄).

The planar test bed disclosed herein greatly mitigates this issue byproviding one or more mechanical couplings capable of establishing aplanar surface orientation as directed by a controller in datacommunication with the mechanical couplings. In brief and as will bediscussed further, the controller receives an object location describinga point on the planar surface from, for example, a camera vision sensingsystem monitoring the planar surface, a communication from a maneuveringtest vehicle, or some other means, and additionally either receives ordetermines a velocity vector describing the state of the object. Inoperation, the “object” in terms of both location and velocity refersgenerally to some test vehicle operating on the planar surface during asimulation. The controller takes the object location and velocity and,based on the position and velocity relative to first location 302,determines a relative acceleration using a function of the form {umlautover (ρ)}_(t)(t)=f(t,ρ,{dot over (ρ)},μ_(t)) or equivalentlya_(R)=f(t,x_(R),v_(R),μ_(t)). The controller then communicates with themechanical couplings to establish an orientation of planar surface Psuch that the local gravity vector projected onto planar surface Pmimics the relative acceleration vector a_(R) determined for thesituation as reported by the system. Generally a_(R) comprises a firstcomponent parallel to the axis x_(p) comprising planar surface P and asecond component comprising the axis y_(p) comprising planar surface P.In this manner, the processor cyclically operates such that the planartest bed provides time-varying relative accelerations on a test vehiclewhich appropriately vary as the test vehicle assumes various statesduring a simulation. For example, relative to FIG. 3, the controller andsystem might operate cyclically to determine positions and velocities asan object assumes the positions indicated as ρ_(S)(t₁), ρ_(S)(t₂),ρ_(S)(t₃), and ρ_(S)(t₄) over some time interval incorporating a t₁, t₂,t₃, and t₄, and provide planar surface orientation such that the localgravity projection accurately reflects the respective relativeaccelerations between those positions and first location 302 that wouldbe expected as the relative spatial orientation on the planar surface Pevolves.

As discussed, the particular embodiment of the planar test bed 100illustrated at FIG. 1 comprises planar table 101, planar surface P, andfirst location 102 on planar surface P. First location 102 is a fixedlocation within the boundary of planar surface P, and planar surface Pfurther comprises an axis x_(p) and an axis y_(p), with the axis x_(p)perpendicular to the axis y_(p). Planar surface P further provides asubstantially flat surface. In an embodiment, the planar surface Pcomprises a contiguous area having a flatness deviation δ_(e) of lessthan 0.01 inches, and in certain embodiments, 0.001 inches. See e.g.,ISO1101, Geometrical product specifications (GPS)—Geometricaltolerancing—Tolerances of form, orientation, location and run-out(2017). In some embodiments, planar surface P has a surface finish of 25micron rms or better, and in other embodiments, planar surface P has alocal flatness not exceeding 5 microns per 40 mm.

Planar test bed 100 further comprises one or more mechanical couplingssuch as mechanical couplings 103, 105, and 107, as earlier indicated.Each mechanical coupling is in mechanical communication with planartable 101 and further in mechanical communication with a supporting leg.For example, at FIG. 1 mechanical coupling 103 is in mechanicalcommunication with both planar table 101 and supporting leg 104,mechanical coupling 105 is in mechanical communication with both planartable 101 and supporting leg 106, and mechanical coupling 107 is inmechanical communication with both planar table 101 and supporting leg108. The mechanical couplings are generally slidably translatable,rotably translatable, or both, such that a mechanical translation of themechanical coupling alters a position vector between some specific,fixed point comprising the planar table and some specific, fixed pointcomprising the supporting leg. For example, at FIG. 1, mechanicalcoupling 103 is mechanically translatable such that a mechanicaltranslation of mechanical coupling 103 alters a position vectorextending from point a on supporting leg 104 and point b on planar table101. Such mechanically translatable couplings are known in the art. In aparticular embodiment, the mechanical couplings comprise leveling wedgesdriven by a motorized element which allows communication from acontroller to direct leveling wedge motion and subsequent mechanicaltranslation of the coupling.

Planar test bed 100 further comprises processor 109 in datacommunication with the one or more mechanical couplings. “Datacommunication” as used here means that a connectivity sufficient toallow data to pass between a first component and a second component ispresent, regardless of whether any such data is presently passing fromthe first component to the second component. For example at FIG. 1,processor 109 is in data communication with mechanical couplings 103,105, and 107 through channels 110, 111, and 112 via hard-wired,wireless, or some other means. As mentioned, processor 109 directsoperations of the one or more mechanical couplings to adjust theorientation of planar table 101 and planar surface P such that a localgravity vector g generates an acceleration parallel to planar surface Pin accordance with a computed function.

Processor 109 is programmed to provide commands to the mechanicalcouplings by executing a series of programmed steps in a generallycyclic fashion. As discussed, the commands generated orient planarsurface P such that the gravity vector g acts through planar surface Pin a manner representing a relative acceleration expected between twobodies in a simulated situation. Generally, processor 109 establishesthe location of one body to the first location 102 of planar surface Pwhile tracking the varying position of a second body over planar surfaceP, such that the observed motion of the second body with respect to thefixed first location 102 models the relative spatial parameters such asthe relative position between the two bodies, the relative velocitybetween the two bodies, and the relative acceleration between the twobodies. For example at FIG. 1, the relative spatial parameters between afirst object and a second object both orbiting a body such as earthmight be represented on planar surface P with the first object anchoredat first location 102, and the relative spatial displacement of thesecond object indicated by the time-varying object locations illustratedas L₁, L₂, L₃, and L₄. Generally at each object location L₁, L₂, L₃, andL₄, processor 109 issues commands to orient planar surface P such thatthe local gravity vector g generates an acceleration vector parallel toplanar surface P that mimics the expected relative acceleration betweenthe two bodies. As discussed, first location 102 may be a fixed point onplanar surface P which remains statically located as depicted, or theposition of first location 102 may vary with time in a manner providedto controller 109, so that for each of L₁, L₂, L₃, and L₄, processor 109determines v_(R) based on the transiently varying first location 102 andan a_(R) based on the x_(R), v_(R), μ_(t). Additionally, it isunderstood that the designation of first location 102 as a first bodyco-orbiting with a second body in the foregoing is exemplary only, andthat first location 102 could be representative of a first body having awide variety of postures relative to the second body. For example, firstlocation 102 could be representative of a first body orbited orapproached by the second body, with L₁, L₂, L₃, and L₄ representing therelative spatial relationship during the orbit or approach. Otherapplicable situations may be envisioned.

In order to provide the necessary orientations of planar surface P,processor 109 is programmed to retrieve a quantified local gravityvector. The quantified local gravity vector provides a value for thelocal gravity expected such as g, and may be retrieved through anyappropriate means, such as retrieval from an embedded memory device,receipt through a manual input, communication with an instrument such asan accelerometer, or other means. Processor 109 is further programmed toreceive an object location at a time t₀, and programmed to treat theobject location at the time t₀ as representative of a location on planarsurface P, such as L₁. Processor 109 may retrieve the object locationfrom any source. In a particular embodiment, processor 109 retrieves theobject location via communication with a camera vision sensing systemsuch as that comprising camera 113 and camera 114.

Processor 109 is further programmed to obtain an object velocity andtreat the object velocity as a directional velocity of the objectlocation received relative to first location 102. Processor 109 mayobtain the object velocity from any source and any appropriate method.Typically processor 109 obtains the object velocity throughcommunication with employed velocity sensing elements. In a particularembodiment, processor 109 obtains the velocity from communications withthe camera vision sensing system comprising cameras 113 and 114. Inanother embodiment, processor 109 is programmed to obtain the velocitythrough communication with instrumentation comprising a test vehiclemoving over planar surface P. Processor 109 treats the object velocityobtained as representative of a velocity at a time t_(i). For example,when processor receives an object location L₁ corresponding to a timet₁, and additionally obtains an object velocity corresponding to theobject location L₁, it treats the corresponding object velocity asrepresentative of the directional velocity of the object at the time t₁with the object generally situated at location L₁.

Processor 109 is further programmed to compute a desired accelerationvector, where the desired acceleration vector is a function of at leastthe first location 102, the object location at a time t_(i) such as L₁at the time t₁, and the corresponding object velocity. In typicalembodiments, processor 109 is further programmed to obtain otherparameters μ_(t), with these parameters being used to modify theacceleration to be applied. Generally, the function defining the desiredacceleration vector may be expressed as a_(R)=f(t,x_(R),v_(R),μ_(t)),where t is the time associated with the object location received, x_(R)is the relative spatial relationship as represented by the objectlocation received, v_(R) is the object velocity obtained, μ_(t) denoteother relevant fixed, or time-varying parameters at time t, and a_(R) isthe desired acceleration vector. Typically x_(R), v_(R) and μ_(t) areexpressed and utilized as vector quantities. The desired accelerationa_(R) may be determined using any appropriate expression, provided theexpression is a function of t, x_(R), v_(R) and μ_(t). In a typicalembodiment, the function models inertial or non-inertial accelerationsdescribing the motion of target and chaser spacecraft during proximitymaneuvers. In other embodiments the function models inertial ornon-inertial accelerations of a spacecraft relative to an objectconsidered stationary. In most cases, appropriate scaling termsnecessary to accurately reduce an expected scenario to the dimension ofthe planar surface P. In a particular embodiment, the function is alinearized equation of relative motion with variables expressed in anLVLH frame. In another embodiment, the linearized equation of relativemotion describes relative motion between two spacecraft orbiting a thirdbody. In some embodiments, the function comprises the orbital rate of atarget spacecraft.

Having determined the desired acceleration vector a_(R), processor 109is programmed to calculate a desired planar surface orientation in whicha vector projection of the quantified local gravity vector onto planarsurface P mimics the desired acceleration vector a_(R). Processor 109calculates the desired planar surface orientation by determining anorientation where, when the planar surface is in the desired planarsurface orientation, the quantified local gravity vector projected ontothe axis x_(p) generates an a_(x1) and the quantified local gravityvector projected onto the axis y_(p) generates an a_(y1), and thedesired acceleration vector projected onto the axis x_(p) generates ana_(x2) and the desired acceleration vector projected onto the axis y_(p)generates an a_(y2), and 0.9≤(a_(x1)/a_(x2))≤1.1 and0.9≤(a_(y1)/a_(y2))≤1.1. In some embodiments, 0.95≤(a_(x1)/a_(x2))≤1.05and 0.95≤(a_(y1)/a_(y2))≤1.05, and in other embodiments,0.99≤(a_(x1)/a_(x2))≤1.01 and 0.99≤(a_(y1)/a_(y2))≤1.01. Statedequivalently, the desired planar surface orientation is an orientationof planar surface P where a_(x1) divided by a_(x2) is greater than orequal to 0.9 and less than or equal to 1.1, in some embodiments greaterthan or equal to 0.95 and less than or equal to 1.05, and in otherembodiments greater than or equal to 0.99 and less than or equal to1.01, and where a_(y1) divided by a_(y2) is greater than or equal to 0.9and less than or equal to 1.1, in some embodiments greater than or equalto 0.95 and less than or equal to 1.05, and in other embodiments greaterthan or equal to 0.99 and less than or equal to 1.01.

As used here, the “desired planar surface orientation” means anorientation of a coordinate system comprising the x_(P) and y_(P) ofplanar surface P with respect to a coordinate system comprising an axisx_(o), an axis y₀, and an axis z_(o), where the axis x_(o), the axis y₀,and the axis z_(o) intersect at an origin O comprising planar table 101,and where the axis z₀ is parallel to the quantified local gravityvector, the axis x₀ is perpendicular to the axis y₀, and the axis y₀ isperpendicular to both the axis z₀ and the axis x₀. Processor 109 maycalculate this desired planar surface orientation using any appropriatemethodology known in the art, provided that the orientation establishesa geometry where at least 0.9≤(a_(x1)/a_(x2))≤1.1 and0.9≤(a_(y1)/a_(y2))≤1.1. An appropriate x_(o)-y_(o)-z_(o) coordinatesystem is illustrated at FIG. 1. Further, “parallel” means that a firstdirection vector is parallel to a first line and a second directionvector is parallel to a second line, and the angle between the firstdirection vector and the second direction vector is less than 5 degrees,preferably less than 2 degrees, and more preferably less than 1 degree.Similarly, when a vector is parallel to a planar surface this means thatthe vector is parallel to a line comprising planar surface. Similarly,“perpendicular” means that the angle between the first direction vectorand the second direction vector is at least 85 degrees, preferably atleast 88 degrees, and more preferably at least 89 degrees.

As an example of an appropriate orientation, FIG. 4 illustrates a planarsurface P of a planar test bed as disclosed with the planar surface Pcomprising the x_(P) and y_(P) axis as previously described. Forillustrative purposes, a z_(P) axis perpendicular to x_(P) and y_(P) isalso included. Additionally illustrated is a location L₅ on planarsurface P, with local gravity vector g extending through L₅. Althoughnot illustrated, the quantified local gravity vector in this scenario issubstantially equal to the local gravity vector g shown. Also shown isorigin O, where origin O is a point comprising the planar test bed. Anaxis y₀, axis x_(o), and axis z_(o) intersect at the axes origin O, withz₀ parallel to the local gravity vector g, the x₀ perpendicular to z₀,and y₀ perpendicular to both z₀ and x₀. At FIG. 4, planar surface P hasan orientation defining an angle θ_(X) between x_(P) and x₀ and an angleθ_(y) between y_(P) and y₀. Lines x′, y′, and z′ are included forreference and are parallel to x₀, y₀, and z₀ respectively. A desiredacceleration a_(R) to be mimicked at location L₅ is additionallyillustrated, where a_(R) is co-planar with x_(P) and y_(P) and intendedto act parallel to planar surface P. The vector projection of thedesired acceleration a_(R) onto x_(P) is illustrated as a_(x2) and thevector projection of the desired acceleration a_(R) onto y_(P) isillustrated as a_(y2).

In the orientation depicted at FIG. 4, the vector projection of thelocal gravity vector g onto x_(P) and y_(P) is indicated by projectionlines L_(X) and L_(Y) respectively. L_(X) projects a component of g ontox_(P) equal to an a_(x1) (not shown for clarity) and L_(Y) projects acomponent of g onto y_(P) equal to an a_(y1) (not shown for clarity). Inthis orientation and with the resulting projection of local gravityvector g, 0.9≤(a_(x1)/a_(x2))≤1.1 and 0.9≤(a_(y1)/a_(y2))≤1.1. In someembodiments 0.95≤(a_(x1)/a_(x2))≤1.05 and 0.95≤(a_(y1)/a_(y2))≤1.05 andin other embodiments 0.99≤(a_(x1)/a_(x2))≤1.01 and0.99≤(a_(y1)/a_(y2))≤1.01. Additionally, it is understood that specificuse of the angles θ_(x) and θ_(y) at FIG. 4 is exemplary only, and thatprocessor 109 may determine and define the necessary orientation in anymanner. For example, processor 109 might track and evaluate variousorientations with reference to rotations about x_(o) and y_(o) andtranslation parallel to z_(o) relative to some fixed point in space suchas 440 using rotation matrices, Euler rotations, unit quaternions, andother means as known in the art.

Having calculated the desired planar surface orientation, processor 109is programmed to issue commands to the one or more mechanical couplingsand direct mechanical translations sufficient to establish planarsurface P in the desired planar surface orientation. In this manner,processor 109 directs an orientation of planar surface P such that thelocal gravity vector g projected onto planar surface P acts to mimic thedesired acceleration vector a_(R) on planar surface P.

Test bed 100 operates cyclically to generate the time-varyinggravity-induced acceleration by receiving a subsequent object locationat a time t_(i), for example L₂ at FIG. 1, and obtaining another objectvelocity, computing another desired acceleration vector, andestablishing planar surface P is an updated orientation, where theupdated orientation mimics the new desired acceleration vector, forexample at L₂. In repeating type fashion, the test bed continues thisprocess so that, for example, a test object traveling over planarsurface P and observed at positions L₁, L₂, L₃, and L₄ will experiencethe relative accelerations that would be expected between the bodyrepresented by the test vehicle and the body represented by firstlocation 102.

The mechanical couplings as discussed are generally slidablytranslatable, rotably translatable, or both, such that a mechanicaltranslation of the one or more mechanical couplings alters a positionvector between some specific, fixed point comprising the planar tableand some specific, fixed point comprising the supporting leg. Inembodiments where the planar test bed 100 comprises the origin O, theaxis x₀, the axis y_(o), and the axis z_(o), when the axis z₀ isparallel to the quantified local gravity vector and the planar surface Pis perpendicular to the axis z_(o), the rotatable, slidable or othermechanical translation of the one or more mechanical couplings providesthree degrees of freedom to planar table 100, with the first degree offreedom a rotation around the axis x₀, the second degree of freedom arotation around the axis y₀, and the third degree of freedom atranslation parallel to the axis z₀. Such mechanical couplings are knownin the art and may be one of or a combination of mechanical couplingsgenerally referred to as leveling wedges, leveling mounts,ball-and-socket joints, hinge joints, slider joints, universal joints,piston joints, and others. Typically the one or more mechanical jointscomprise a motorized element in data communication and receivingcommands from processor 109, and the motorized element generates motiveforce acting on and repositioning the mechanical joint in order toprovide the desired orientation of planar surface P. In someembodiments, the motorized elements may communicate back to processor109 in order to report a current mechanical posture of the mechanicalcoupling.

In some embodiments, planar table 101 further comprises one or moreinclinometers in data communication with processor 109 and providingindications to processor 109 describing a current orientation of planarsurface P. For example, in one embodiment, planar table 101 comprisesinclinometers 118 and 119 where inclinometer 118 has a measurement axisparallel to axis x_(p) of planar surface P and provides an inclinationrelative to axis x_(o), and inclinometer 119 has a measurement axisparallel to axis y_(p) of planar surface P and provides inclinationrelative to axis y_(o). Such inclinometers are known in the art andgenerally determine inclinations using a local gravity vector such as g.As is understood, a single biaxial inclinometer may also be employed. Inan embodiment, planar table 101 comprises one or more inclinometers indata communication with processor 109, and processor 109 is furtherprogrammed to receive a first inclination and a second inclination,treat the first inclination and second inclination as indicative of thecurrent orientation of the axes x_(p) and y_(p) relative to at least theaxis x_(o) and the axis y_(o), and determine a current planar surfaceorientation of planar surface P utilizing the first inclination and thesecond inclination.

In other embodiments, planar test bed 100 comprises one or more andtypically a plurality of laser distance meters in data communicationwith processor 109 and providing distance measures to processor 109 inorder to describe a current orientation of planar surface P. Forexample, at FIG. 1, planar table 101 comprises laser distance meters120, 121, and 122 at three positions on planar table 101 with eachhaving a measurement axis sufficiently aligned to determine a distanceparallel to the axis z_(o) at each location. In an embodiment, planartable 101 comprises the plurality of laser distance meters in datacommunication with processor 109, and processor 109 is furtherprogrammed to receive one or more distances and treat the one or moredistances as indicative of a distance at a fixed location on planartable 101 and parallel to the quantified local gravity vector, anddetermine a current planar surface orientation of planar surface Putilizing the one or more distances. The current planar surfaceorientations described here may be used for a variety of purposes,including serving as feedback to processor 109 while enroute to a new,updated orientation.

In certain embodiments, processor 109 obtains an object position viacommunication with a camera vision sensing system such as thatcomprising camera 113, camera 114, and image processor 115. Imageprocessor 115 may be a processor wholly distinct from processor 109 andin data communication with processor 109 via 123 as illustrated, orimage processor 115 may be a set of instructions residing withinprocessor 109. In these embodiments, cameras 113 and 114 are orientedsuch that the field-of-view (FOV) of each camera includes at least someportion of planar surface P. In other embodiments, processor 109 orimage processor 115 obtains an object velocity using two or more imagesreceived from one or more of the cameras based on a displacementdepicted and an elapsed time.

The cyclic nature of the operations of processor 109 is depictedgenerally at FIG. 5. At FIG. 5, operations commence at BEGIN. At 526,the processor retrieves the quantified local gravity vector g_(v)through an appropriate means, such as retrieval from an embedded memorydevice, receipt through a manual input, communication with an instrumentsuch as an accelerometer, or other means. At 527, the processorcommences determination of a desired acceleration for a time t_(i). At528, the processor receives an object location x_(R) at the time t_(i)and treats the object location x_(R) as representative of a location onplanar surface P. As discussed, processor 109 may retrieve the objectlocation from any source. In an embodiment, a camera vision sensingsystem provides the object location x_(R). At 529, the processor obtainsan object velocity v_(R) and treats the object velocity as a directionalvelocity of the object location received, relative to first location 102of planar surface P. Processor 109 may obtain the object velocity fromany source, for example, from the camera vision sensing system orthrough communication with on-board sensors comprising a test vehicle.

At 530, processor 109 computes a desired acceleration a_(R) for the timet_(i) using a function comprising time t, the object location x_(R)retrieved, the object velocity v_(R) obtained, and typically therequired relevant orbital, environmental, and scaling parameters μ_(t).In a typical embodiment, the desired acceleration is determined with afunction a_(R)=f(t,x_(R),v_(R),μ_(t)). When applicable, the processormay receive the required parameters μ_(t) from any source. Some of theparameters μ_(t) utilized in the function results from communicationswith a test vehicle operating on planar surface P.

At 531, processor 109 calculates the desired planar surface orientationto reproduce the desired acceleration a_(R) on planar surface P, bydetermining an orientation where a vector projection of the quantifiedlocal gravity vector g_(v) onto a plane comprising the x_(p) and y_(p)axes of the planar surface P sufficiently mimics the desiredacceleration a_(R). The appropriate orientation is an orientation wherethe quantified local gravity vector g_(v) projected onto the axis x_(p)generates an acceleration a_(x1) and the desired acceleration vectorprojected onto the axis x_(p) generates an a_(x2), with0.9≤(a_(x1)/a_(x2))≤1.1, and also where the quantified local gravityvector projected onto the axis y_(p) generates an a_(y1) and the desiredacceleration vector projected onto the axis y_(p) generates an a_(y2),with 0.9≤(a_(y1)/a_(y2))≤1.1.

At 532, the processor directs the one or more mechanical couplings suchas M₁, M₂, and M₃ to mechanically translate and establish planar surfaceP in the desired planar surface orientation.

At 533, the processor determines if continued operations are required,and if so, increments t_(i) at 534 to establish a new t_(i), and repeatsat least steps 527-533. At 533, processor 109 may use any indication todetermine continuation, such as a maximum t_(i), a particular relativeposition x_(R) received, a manual input, or any other criteria by whichcontinuation of the process may be affirmed. Further, it is understoodthat the duration of the time interval Δt at FIG. 5 may vary from cycleto cycle and need not be constant.

In a particular embodiment, the rigid flat table (e.g. a granitemonolith) is mounted on top of three height-adjustable pedestals. Bychanging the height of the pedestals the orientation of the table's flatsurface with respect to the local gravity is adjusted. When the tabledeviates from its leveled orientation, a gravitational acceleration isimparted to the test vehicles. The actuators that regulate the height ofthe pedestals are connected to a central controller (i.e. a computer)that coordinates their action so that the desired table orientation(i.e. acceleration) is obtained. Two high precision inclinometers (orone two-axis inclinometer) measure the orientation of the operatingsurface with respect to the local gravity and provide feedback to thecontroller. By measuring the height of three points of the flat tablethe orientation of the table can also be determined. Three highprecision laser distance meters can be used for this task.

Assume, without loss of generality, that the three pedestals to surfacecontact point positions are known with respect to a fixed referenceframe with a Cartesian Coordinate System having the z-axis aligned withthe local gravity. Let the vector from the origin of this referenceframe to the three supports be denoted by p_(i) with i=1, 2, 3. A vectorn that is normal to the table's surface can be easily computed asfollows.n=(p ₃ −p ₁)×(p ₂ −p ₁).

Define, without a loss in generality, a reference frame that is attachedto the test bed flat table with its z⁰-axis perpendicular to the surface(i.e. along n). If {circumflex over (n)} denotes the normalized n vectorand g=9.81 m/s² denotes local surface gravity magnitude, then theacceleration imparted on a test vehicle floating on the flat table framecan be determined as follows.a _(x) ′=g{circumflex over (n)} _(x)a _(y) ′=g{circumflex over (n)} _(y)

The orientation of the flat table can be time-varying in order to imparta time-varying acceleration a_(R)(t). In general, a model f can be usedto determine the required acceleration a_(R) given the current time t,as well as the position x_(R) and velocity v_(R) based on the testvehicle. Other time-varying parameters μ_(t)(t) may also be used todrive the f model.a _(R) =f(t,x _(R) ,v _(R),μ_(t))

As the f model may depend on the state of the test vehicle (x_(R) andv_(R)) this state must be known by the controller. Different methods canbe used to determine the state of the vehicles (for example a motioncapture system or state estimation provided by the test vehicle(s)). Awireless link between the test vehicle and the controller can be used bythe vehicle to transmit some of the μ_(t) parameters that may be neededby the f model. The controller can also inform the test vehicle of theflat table orientation and the resulting acceleration a_(R).

A particularly suitable disposition to actuate the height-adjustablepedestals is a leveling wedge actuated by an electric motor. FIG. 6shows a notional representation of this particular embodiment of aheight-adjustable pedestal. It is customary to mount granite monolithsfor planar air bearings on top of three pedestals. Leveling wedges arethen used to manually adjust the height of pedestals in order to ensurethe horizontality of the flat table once installed (i.e. compensatingfor uneven foundations). At FIG. 6, the leveling wedges are motorizedwith an electric motor. The leveling wedges are then actuated to achievethe required flat table orientation. A setting screw is used to regulatethe leveling wedge's height. An electric motor coupled to a highreduction gear can be used to actuate this screw and automate theheight-adjustment procedure. FIG. 6 illustrates the scenario depicting atest vehicle and planar table, with a leveling wedge mount driven by anelectric motor, and a motor driver in data communication with acontroller.

The central controller is used to coordinate the electric motors thatactuate the wedge leveling mounts. Although an open-loop scheme could beused to control the three motors, two high precision inclinometers orthree laser distance meters can be used to provide feedback to thecontroller. An overview of all the elements in an embodiment of thesystem is shown in FIG. 7, illustrating a test vehicle with air bearingsatop a planar table, with the planar table comprising one or moreinclinometers. Supporting legs 751, 752, and 753 are each in mechanicalcommunication with a leveling wedge as illustrated, with each levelingwedge driven by a motor and motor driver. Each motor driver is in datacommunication with a controller comprising a digital processor (PC). ThePC further comprises a model for the computation of a relativeacceleration a_(x,y). A laser distance meter is also illustrated in datacommunication with the controller.

An alternative realization uses a linear motor in conjunction with oneor more supporting legs as shown in FIG. 8. FIG. 8 illustrates a planartable comprising a planar surface. A mechanical coupling comprising acaptive ball joint, power screw, and bevel gear is driven by an electricmotor and in mechanical communication with the planar table and apedestal. A motor driver is in communication with the electric motor andin data communication with a controller.

Additional relevant information on the operations of a particularembodiment of the planar test bed is presented below.

Description of a Specific Embodiment

The invention disclosed provides a floating spacecraft simulator thatcomprises a flat table that can dynamically change its orientation withrespect to the local gravity and thus impart a time-varying accelerationto the test vehicles. This capability can be used to impart arbitrarytime-varying accelerations to the test vehicles without using theiractuators, thus greatly extending the applicability of planar airbearing test beds.

The non-inertial accelerations that appear when studying the motion ofspacecraft during proximity maneuvers provide a paradigmatic example ofthe utility of this concept. By tilting the flat table thesenon-inertial accelerations can be imparted to the test vehicles. Asthese accelerations are dependent on the relative state between thespacecraft and the origin of the non-inertial frame (e.g. see theHill-Clohessy-Wiltshire equations) the tilt of the flat table mustconstantly change to impart the correct acceleration (magnitude anddirection).

A spacecraft orbiting a small and irregular planetary body is anotherexample where a spacecraft would be subject to a time-varying residualacceleration that can be recreated by continuously tilting the flattable. In some other scenarios, other orbital perturbations cannot beneglected either (e.g. aerodynamic forces or solar radiation pressureperturbations). By tilting the flat table, the effects of thesetime-varying perturbations can also be recreated. By tilting the flattable all the vehicles floating on its top experience the sameacceleration.

Unfortunately air-bearing test beds have operating tables limiteddimensions and in most instances the considered problems span muchlarger distances. To accommodate these test scenarios, generic scalingof the system dynamics allows to replication on small operatingsurfaces.

Scaling of the Problem Dynamics:

The Buckingham's Pi theorem is utilized to derive the scaling laws. Thistheorem states that for a system described by p variables which involveq independent dimensional units, there exists p-q dimensionless 7Lparameters that must remain invariant during the scaling process inorder to guarantee dynamic similitude. The following vectorialrelationships can be used to fully describe the test vehicle'stranslational dynamics, with F denoting the resultant force acting onthe vehicle, m its mass, t time, and a, v, and r denoting the vehicle'sacceleration, velocity and position respectively

$\begin{matrix}{F = {ma}} & \left( {1a} \right) \\{a = \frac{d\; v}{d\; t}} & \left( {1b} \right) \\{v = \frac{d\; r}{d\; t}} & \left( {1c} \right)\end{matrix}$

As the test vehicle is moving on a plane, Eq. 2 can be decomposed intotwo systems of scalar equations.

$\begin{matrix}{F_{i} = \begin{matrix}{ma}_{i} & {{i = 1},2}\end{matrix}} & \left( {2a} \right) \\{a_{i} = \begin{matrix}\frac{d\; v_{i}}{d\; t} & {{i = 1},2}\end{matrix}} & \left( {2b} \right) \\{v_{i} = \begin{matrix}\frac{d\; r_{i}}{d\; t} & {{i = 1},2}\end{matrix}} & \left( {2c} \right)\end{matrix}$

Focusing in any of these two systems it is immediate to observe that itcontains p=6 relevant variables (F_(i), m, a_(i), v_(i), r_(i), t) whichinvolve q=3 independent dimensional units (mass M, length L, and timeT). According to Buckingham's Pi theorem, there are p−q=3 independent πdimensionless parameters.

A systematic method to derive these dimensionless parameters is toselect q “core” variables from the p set to represent the q independentdimensional units. For example, we can select m for the mass M, x_(i)for distance L, and t for time T. The p−q π dimensionless parameters canbe found by finding the combination of the “core” variables that whenmultiplied with each of the remaining p variables produces adimensionless parameter. This problem can be formulated as finding the μcoefficients that make the following π parameters dimensionless.π₁ =t ^(μ1,1) r _(i) ^(μ2,1) m ^(μ3,1) v _(i)  (3a)π₂ =t ^(μ1,2) r _(i) ^(μ2,2) m ^(μ3,2) a _(i)  (3b)π₃ =t ^(μ1,3) r _(i) ^(μ2,3) m ^(μ3,3) F _(i)  (3c)

It is then straightforward to find the p coefficients and thus the πdimensionless parameters that must be kept constant in order to preservethe dynamic similitude are the following.

$\begin{matrix}{\pi_{1} = \frac{{tv}_{i}}{r_{i}}} & \left( {4a} \right) \\{\pi_{2} = \;\frac{t^{2}a_{i}}{r_{i}}} & \left( {4b} \right) \\{\pi_{3} = \;\frac{t^{2}F_{i}}{{mr}_{i}}} & \left( {4c} \right)\end{matrix}$

If a variable on the original problem is denoted by Q its scaled versionwill be denoted by Q⁰. We can then define its scaling factor as follows.

$\begin{matrix}{\lambda_{Q} = \frac{Q}{Q^{\prime}}} & (5)\end{matrix}$

By imposing the invariance of the Eq. 4 dimensionless parameters thefollowing scaling laws between the different scaling factors can bederived.λ_(ri)=λ_(t)λ_(vi)  (6a)λ_(ri)=λ_(2t)λ_(ai)  (6b)λ_(m)λ_(ri)=λ_(2t)λ_(Fi)  (6c)

For a one-dimensional case, there are 6 scaling parameters λ_(Q) to setand only 3 linearly independent nonlinear equations.

These scaling laws can be utilized to greatly expand the applicabilityof the test bed. Assume emulation of the Apollo Moon landings on thePOSEIDYN test bed. See Zappulla II, R., Virgili-Llop, J., Zagaris, C.,Park, H., and Romano, M., “Dynamic Air-Bearing Hardware-in-the-LoopTestbed to Experimentally Evaluate Autonomous Spacecraft ProximityManeuvers,” Journal of Spacecraft and Rockets, in press, 2017. The massof the Lunar Module was approximately 15000 kg and it maximum thrust of45 kN. The POSEIDYN test vehicle's mass is 10 kg and their maximumthrust is 0.3 N. Therefore, select λ_(m)=10 and λ_(F)=500/3. To recreatethe last 300 m of landing phase in the 4 m granite monolith thenλ_(x)=10. It thus follows that λ_(t)=0.87, λ_(a)=100, and λ_(v)=86. TheMoon's surface gravity of g=1.625 m/s² will have to be simulated in thetest bed as a g′=0.01625 m/s², which corresponds to tilting the table0.095′. As the scaled Moon acceleration g′ is several orders ofmagnitude larger than the 19 μg of residual acceleration on the POSEIDYNtest bed (equivalent to a surface tilt of 0.0011′), the scaled Moongravity could be recreated without being significantly impacted by thetest bed residual acceleration. Another interesting point to note aboutthis example is that the maneuver time on the air bearing test bed willonly be 13% slower than the real landing maneuver. By forcing the flightcomputer to be idle for 13% of the time, the guidance navigation andcontrol software can be tested with equivalent real-time conditions.

An upcoming mission might attempt to perform a touch-and go maneuver onthe near-Earth asteroid Bennu. With the appropriate scaling, thismaneuver can also be practiced on the POSEIDYN test bed. The spacecraftmass is approximately 1500 kg and uses two 0.08 N thrusters. The finaltouch-and-go maneuver occurs from an altitude of 125 m. The resultingscaling is λ_(m)=150, λ_(F)=0.53, and λ_(x)=31.2, λ_(t)=93.8,λ_(a)=0.0036, and λ_(v)=0.33. The surface gravity of Bennu isg_(B)≈2.1×10⁻⁵ m/s², which is well below the residual acceleration ofthe POSEIDYN test bed. When the scaling is factored in then g_(B)⁰≈0.0059 m/s² (corresponding with a tilt of 0.034°), which isapproximately 30 times larger than the test bed residual acceleration.In this case the maneuver in the test bed occurs much faster than thereal maneuver (λ_(t) times faster) and thus if the guidance and controlsoftware can meet the more stringent time requirements in the test bedit will have time to spare once in space.

This analysis can be easily extended to the two-dimensional case. Inthat case there are p=9 relevant variables(F_(1,2),m,a_(1,2),v_(1,2),r_(1,2),t) and still q=3 independentdimensional units. Therefore p−q=6 independent 7 r dimensionlessparameters can be defined.

$\begin{matrix}\begin{matrix}{\pi_{1} = \frac{{tv}_{1}}{r_{1}}} & {\pi_{4} = \frac{{tv}_{2}}{r_{2}}}\end{matrix} & \left( {7a} \right) \\\begin{matrix}{\pi_{2} = \;\frac{t^{2}a_{1}}{r_{1}}} & {\pi_{5} = \;\frac{t^{2}a_{2}}{r_{2}}}\end{matrix} & \left( {7b} \right) \\\begin{matrix}{\pi_{3} = \;\frac{t^{2}F_{1}}{{mr}_{1}}} & {\pi_{6} = \;\frac{t^{2}F_{i}}{{mr}_{2}}}\end{matrix} & \left( {7c} \right)\end{matrix}$

The scaling factor relationships also double when the additionaldimension is introduced.λ_(r1)=λ_(t)λ_(v1) λ_(r2)=λ_(t)λ_(v2)  (8a)λ_(r1)=λ_(2t)λ_(a1) λ_(r2)=λ_(2t)λ_(a2)  (8b)λ_(m)λ_(r1)=λ_(2t)λ_(F1) λ_(m)λ_(r2)=λ_(2t)λ_(F2)  (8c)

These scaling relationships show that the scaling in the two differentdimensions can be different. In the case of the planetary landings thiscan be used to scale the along track landing dimension, which may not beof the same magnitude that the maneuver's vertical dimension.

The scaling can also be used to recreate the relative orbital dynamics.Assume a standard case of a spacecraft operating in close proximity of aResident Space Object (RSO) that is in a circular orbit around theEarth. The relative motion of the spacecraft in relation to the RSO canbe approximated, in the local-vertical local-horizontal frame, by thewell known Hill-Clohessy-Wiltshire equations shown in Eq. 9.

$\begin{matrix}{a_{x} = {{{- 2}{nv}_{y}} + \frac{F_{x}}{m}}} & \left( {9a} \right) \\{a_{y} = {{3n^{2}r_{y}} + {2{nv}_{x}} + \frac{F_{y}}{m}}} & \left( {9b} \right) \\{a_{z} = {{{- n^{2}}r_{z}} + \frac{F_{z}}{m}}} & \left( {9c} \right)\end{matrix}$

In Eq. 9, the x direction is along the RSO orbital velocity (V-bar), ypoints towards the center of the Earth (R-bar) and z is perpendicular tothe orbit plane. The mean motion of the orbit is denoted by n and can becomputed by Eq. 10, where μ is the Earth's and a denotes the orbit'ssemi-major axis. It is worth pointing out at this point that therelationship between the orbit's period P and the mean motion can beexpressed as in Eq. 11 and then it is straightforward to derive therelationship between λ_(n) and λ_(t).

$\begin{matrix}{n = \sqrt{\frac{\mu}{a^{3}}}} & (10) \\{P = \frac{2\pi}{n}} & (11) \\{\lambda_{n} = \frac{1}{\lambda_{i}}} & (12)\end{matrix}$

In this approximation the relative motion occurring out of the orbit'splane is independent from the in-plane motion and can be then controlledindependently. For a planar test bed it may be more interesting tosimulate the in-plane relative orbital dynamics. By appropriatelyscaling the problem and by dynamically tilting the test bed's operatingsurface the relative orbital dynamics effects can be emulated.

To illustrate the procedure on how to determine the scale factors andthe operating surface tilt time history an example is here provided.Assume that the spacecraft is simply in a relative periodic orbit aroundthe RSO. If there are no external forces on the spacecraft F_(x,y,z)=0the in-plane Clohessy-Wiltshire equations have the following closed formsolution.

$\begin{matrix}{{r_{x}(t)} = {r_{x\; 0} + {6\left( {{\sin\mspace{11mu}{nt}} - {nt}} \right)r_{y\; 0}} - {\frac{2}{n}\left( {1 - {\cos\;{nt}}} \right)v_{y\; 0}} + {\frac{{4\;\sin\;{nt}} - {3\;{nt}}}{n}v_{x\; 0}}}} & \left( {13a} \right) \\{\mspace{79mu}{{r_{y}(t)} = {{\left( {4 - {3\cos\;{nt}}} \right)r_{y\; 0}} + {\frac{\sin\;{nt}}{n}v_{y\; 0}} + {\frac{2}{n}\left( {1 - {\cos\;{nt}}} \right)v_{x\; 0}}}}} & \left( {13b} \right)\end{matrix}$

By selecting the Eq. 14 initial conditions, the periodic orbit describedin Eq. 15 is obtained.

$\begin{matrix}\begin{matrix}{{r_{x}(0)} = \frac{2v_{0}}{n}} & {{v_{x}(0)} = 0}\end{matrix} & \left( {14a} \right) \\\begin{matrix}{{r_{y}(0)} = 0} & {{v_{y}(0)} = 0}\end{matrix} & \left( {14b} \right) \\{{r_{x}(t)} = {\frac{2\;\cos\;{nt}}{n}v_{0}}} & \left( {15a} \right) \\{{r_{y}(t)} = {\frac{\sin\;{nt}}{n}v_{0}}} & \left( {15b} \right)\end{matrix}$

The velocities and accelerations can be computed as follows.v _(x)(t)=−2v ₀ sin nt a _(x)(t)=−2nv ₀ cos nt  (16a)v _(y)(t)=v ₀ cos nt a _(y)(t)=−nv ₀ sin nt  (16b)

Assume simulating a circumnavigation of the International Space Station(ISS), so let x₀=450 m so that the m is bigger than the ISS keep-outzone. As the POSEIDYN granite table is 4-by-4 meters let λ_(rx)=62.50and λ_(ry)=31.25. The ISS orbital altitude is approximately 400 km whichmakes its period around 92 minutes. To limit the circumnavigation toaround 4 minutes then λ_(t)=23.10. From these scaling factors it followsλ_(ax)=0.21 and λ_(qy)=0.11. In this circumnavigation case, the relativeorbital dynamics acceleration ranges from 0 to max(a_(x))=2nv₀ andmax(a_(y))=nv₀. When the λ_(ax,y) scaling is applied these maximumaccelerations correspond to a tilt angle of 0.016′ (in eitherdirection).

Another interesting aspect may be to compute which is the percentage ofthis relative orbit where the relative accelerations fall below acertain threshold (so that they cannot be accurately reproduced in thetest bed).a′ _(x) ≤a  (17)a′ _(y) ≤a  (18)

Define that threshold as 2 times the POSEIDYN residual accelerationa_(tres)=38 μg. In this case study that would occur 17% of the orbit(around the vertex and co-vertex of the ellipse).

It is interesting to note that the scaled acceleration is independent ofthe size of the relative orbit. It only depends in the size of the testbed operating surface and the scaled orbit period (see Eq. 19).

$\begin{matrix}{a_{x}^{\prime} = {{- \frac{2{nv}_{0}\sin\;{nt}}{\lambda_{a_{x}}}} = {{{- n^{2}}r_{0\; x}\frac{\lambda_{t}^{2}}{\lambda_{r_{x}}}\sin\;{nt}} = {{- {r_{0\; x}^{\prime}\left( \frac{2\pi}{P^{\prime}} \right)}^{2}}\sin\; n^{\prime}t^{\prime}}}}} & (19)\end{matrix}$

This similitude equivalence far from being specific for acircumnavigation case is shared with the general relative orbitaldynamics accelerations, making the scaling very simple and convenient. Amaneuver performed in the test bed shares the dynamic simulated with thesame maneuver regardless of the RSO orbiting altitude and relativedistance.i _(x)′=−2n′v _(y)′  (20a)(a _(y)′=3n′ ² e _(y)′+2n′v _(x)′  (20a)i _(z) ′=−n′ ² r _(z)′  (20c)

These relative orbital dynamics accelerations are generated by tiltingthe operating surface. The remaining acceleration from the onboardthrusters can be scaled using Eq. 6c, and in this case the operatingaltitude and maneuver distance will have an effect on the scaling.

The FSS has a mass of 10 kg. Assume that we want to simulate the Cygnusspacecraft that approximately has a berthing mass of 5300 kg, thenλ_(m)=530. That leaves λ_(Fx)=113.7 and λ_(Fy)=56.9. The FSS has amaximum thrust per axis of 0.3 N which when scaled would represent amaximum thrust of F_(x)=34 N and F_(y)=17 N which are smaller butsimilar to the Cygnus capabilities.

Although the Clohessy-Wiltshire equations are a simplification of therelative orbital dynamics they can be used to quickly derive the scalingfactors. For a more generic case, use where a_(R) as the relativeorbital dynamics acceleration,

$\frac{F}{m}$to represent the acceleration induced by the control forces (i.e. by thethrusters) and a_(others) to denote the acceleration by otherperturbations (e.g. solar radiation pressure or drag).

$a = {a_{R} + \frac{F}{m} + a_{others}}$

Regardless of how the relative orbital dynamics a_(R) and otherperturbations. a_(others) are computed, these accelerations arerecreated by tilting the planar air bearing operating surface.

Thus provided here is planar test bed comprising a planar table having aplanar surface for the simulation or relative accelerations between afirst location on the planar surface and an object location. The planartest bed comprises one or more mechanical couplings with each inmechanically communication with the planar table a supporting leg, withthe mechanical couplings mechanically translatable to provide threedegrees of freedom to the planar table. A processor in datacommunication with the mechanical couplings and issues commands to themechanical couplings to alter the spatial orientation of the planartable in a transient process, in order to allow a local gravity vector gto mimic an anticipated relative acceleration between two bodies in asimulated environment, typically using a functiona_(R)=f(t,x_(R),v_(R),μ_(t)). The planar test bed operates in a cyclicmanner so that desired accelerations and planar orientations are updatedas an object transits over the planar surface. In a particularembodiment, the system further comprises a camera vision sensing systemproviding at least positional data to the processor during the cyclicprocess.

Accordingly, this description provides exemplary embodiments of thepresent invention. The scope of the present invention is not limited bythese exemplary embodiments. Numerous variations, whether explicitlyprovided for by the specification or implied by the specification ornot, may be implemented by one of skill in the art in view of thisdisclosure.

It is to be understood that the above-described arrangements are onlyillustrative of the application of the principles of the presentinvention and it is not intended to be exhaustive or limit the inventionto the precise form disclosed. Numerous modifications and alternativearrangements may be devised by those skilled in the art in light of theabove teachings without departing from the spirit and scope of thepresent invention. It is intended that the scope of the invention bedefined by the claims appended hereto.

In addition, the previously described versions of the present inventionhave many advantages, including but not limited to those describedabove. However, the invention does not require that all advantages andaspects be incorporated into every embodiment of the present invention.

All publications and patent documents cited in this application areincorporated by reference in their entirety for all purposes to the sameextent as if each individual publication or patent document were soindividually denoted.

What is claimed is:
 1. A planar test bed for creating time-varyinggravity-induced accelerations comprising: a planar table, the planartable comprising a planar surface and the planar surface comprising afirst location, and where the planar surface comprises an axis x_(p) andan axis y_(p) where the axis x_(p) is perpendicular to the axis y_(p);one or more mechanical couplings, where each mechanical coupling is inmechanical communication with the planar table and a supporting leg, andwhere the each mechanical coupling is mechanically translatable, where amechanical translation of the each mechanical coupling alters a positionvector between a specific point on the planar table and a specific pointon the supporting leg; and a processor in data communication with theone or more mechanical couplings where the processor is programmed toperform steps comprising: retrieving a quantified local gravity vector;and performing a cyclic operation to create the time-varyinggravity-induced accelerations by: receiving an object location at a timet₀, where the object location at the time t₀ is a location on the planarsurface; obtaining an object velocity, where the object velocity is arelative velocity between the object location at the time t₀ and thefirst location comprising the planar surface; computing a desiredacceleration vector a_(R), where the desired acceleration vector a_(R)is a function of at least the first location comprising the planarsurface, the object location at the time t₀, and the object velocity;calculating a desired planar surface orientation comprising: projectingthe quantified local gravity vector onto the axis x_(p) to generate ana_(x1), projecting the quantified local gravity vector onto the axisy_(p) to generate an a_(y1), projecting the desired acceleration vectora_(R) onto the axis x_(p) to generate an a_(x2), and projecting thedesired acceleration vector a_(R) onto the axis y_(p) to generate ana_(y2), and wherein 0.9≤(a_(x1)/a_(x2))≤1.1 and 0.9≤(a_(y1)/a_(y2))≤1.1;mimicking the desired acceleration vector a_(R) on the planar surface,by communicating with one or more mechanical couplings and directing theone or more mechanical couplings to mechanically translate and establishthe planar surface in the desired planar surface orientation; andreceiving a subsequent object location at a time t_(i), where the timet_(i) is later than the time t₀, and repeating the obtaining the objectvelocity step, the computing the desired acceleration vector a_(R) step,the calculating the desired planar surface orientation step, and themimicking the desired acceleration vector a_(R) on the planar surfacestep using the subsequent object location at the time t_(j) as theobject location at the time t₀, thereby creating the time-varyinggravity-induced acceleration.
 2. The planar test bed of claim 1 wherethe planar surface comprises an axes origin, and where an axis x₀, andaxis y_(o), and an axis z_(o) intersect at the axes origin, where theaxis x₀ is perpendicular to the axis y₀ and the axis z₀ is perpendicularto the axis y₀ and the axis x₀, and where when the axis z₀ is parallelto the quantified local gravity vector and the planar surface isperpendicular to the axis z_(o), the mechanical translation of the oneor more mechanical couplings provides three degrees of freedom to theplanar table, where the first degree of freedom is a rotation around theaxis x₀, the second degree of freedom is a rotation around the axis y₀,and the third degree of freedom is a translation parallel to the axisz₀.
 3. The planar test bed of claim 2 where the desired planar surfaceorientation calculated by the processor defines an angle θ_(x) betweenthe axis x_(P) and the axis x_(o) and an angle θ_(y) between the axisy_(P) and the axis y_(o), and at least one of the angle θ_(x) and theangle θ_(z) is greater than zero.
 4. The planar test bed of claim 1where the processor is programmed to obtain the object velocity byperforming steps comprising: collecting a plurality of object locationswhere the plurality of object locations comprises the object location atthe time t₀ and the subsequent object location at the time t_(i); andcalculating the object velocity using the plurality of object locations,the time t_(o), and the time t_(i).
 5. The planar test bed of claim 4further comprising a motion sensing system comprising a camera, wherethe camera has a field of view and where the planar table comprises atleast some portion of the field of view, and where the camera is in datacommunication with the processor, and where the processor is furtherprogrammed to perform steps comprising: receiving a plurality of imagesfrom the camera where the plurality of images comprises an image for thetime t_(o) and an image for the time t_(i); determining the objectlocation at the time t₀ using at least the image for the time t_(o) anddetermining the subsequent object location at the time t_(i) using atleast the image for the time t_(i).
 6. The planar test bed of claim 4further comprising an object in data communication with the processor,where the processor is further programmed to receive the object locationat the time t₀ and the subsequent object location at the time t_(i) fromthe object.
 7. The planar test bed of claim 1 where the planar tablefurther comprises one or more inclinometers, where the one or moreinclinometers have a first measurement axis parallel to the axis x_(p)and a second measurement axis parallel to the axis y_(p), and the one ormore inclinometers in data communication with the processor, and wherethe processor is further programmed to receive a first inclination fromthe one or more inclinometers and a second inclination from the one ormore inclinometers and determine a current planar surface orientationutilizing the first inclination and the second inclination.
 8. Theplanar test bed of claim 1 where the planar table further comprises oneor more distance meters in data communication with the processor, andwhere the processor is further programmed to receive a one or moredistance measures from the one or more distance meters and determine acurrent planar surface orientation utilizing the one or more distancemeasures.
 9. A system for creating time-varying gravity-inducedacceleration on a planar surface, the system comprising: a planar table,the planar table comprising a planar surface and the planar surfacecomprising a first location, and the planar surface comprising an axisx_(p) and an axis y_(p) where the axis x_(p) is perpendicular to theaxis y_(p), and a local gravity vector acting on the planar table; anobject in contact with the planar surface and moving across the planarsurface, and the object residing at an object location, where the objectlocation is a location of the object on the planar surface relative tothe first location comprising the planar surface; one or more mechanicalcouplings where each mechanical coupling is in mechanical communicationwith the planar table and a supporting leg, and where the eachmechanical coupling is mechanically translatable, where a mechanicaltranslation of the each mechanical coupling alters a position vectorbetween a specific point on the planar table and a specific point on thesupporting leg; a processor in data communication with the one or moremechanical couplings and performing steps comprising: receiving theobject location; obtaining an object velocity, where the object velocityis a relative velocity between the object and the first locationcomprising the planar surface; computing a desired acceleration vectora_(R), where the desired acceleration vector a_(R) is a function of atleast the first location comprising the planar surface, the objectlocation, and the object velocity; calculating a desired orientation ofthe planar surface comprising: projecting the local gravity vector ontothe axis x_(p) to generate an a_(x1), projecting the local gravityvector onto the axis y_(p) to generate an a_(y1), projecting the desiredacceleration vector a_(R) onto the axis x_(p) to generate an a_(x2), andprojecting the desired acceleration vector a_(R) onto the axis y_(p) togenerate an a_(y2), wherein 0.9≤(a_(x1)/a_(x2))≤1.1 and0.9≤(a_(y1)/a_(y2))≤1.1; communicating the desired orientation of theplanar surface to one or more mechanical couplings, where eachmechanical coupling is in mechanical communication with the planar tableand a supporting leg, and where the each mechanical coupling ismechanically translatable, where a mechanical translation of the eachmechanical coupling alters a position vector between a specific point onthe planar table and a specific point on the supporting leg; the one ormore mechanical couplings mechanically translating, and the mechanicaltranslations of the one or more mechanical couplings establishing theplanar surface in the desired orientation of the planar surface; and theobject transiting over the planar surface from the object location to asubsequent object location, where the subsequent object location isanother location of the object on the planar surface relative to thefirst location comprising the planar surface, and the system repeatingthe receiving the object location step, the obtaining the objectvelocity step, the computing the desired acceleration vector a_(R) step,the calculating the desired orientation of the planar surface step, thecommunicating the desired orientation of the planar surface to the oneor more mechanical couplings step, and the one or more mechanicalcouplings mechanically translating step using the subsequent objectlocation as the object location, thereby creating the time-varyinggravity-induced acceleration.
 10. The system of claim 9 where the planartable comprises an origin O, and where an axis z_(o) is parallel to thelocal gravity vector and intersects the origin O, and where an axis x₀is perpendicular to the axis z₀ and an axis y₀ is perpendicular to boththe axis z₀ and the axis x₀, and where the desired orientation of theplanar surface defines an angle θ_(x) between the axis x_(P) and theaxis x_(o) and an angle θ_(y) between the axis y_(P) and the axis y_(o),and further comprising the one or more mechanical couplings mechanicallytranslating and establishing the angle θ_(x) and the angle θ_(y). 11.The system of claim 10 further comprising: a motion sensing systemcomprising a camera, where the camera has a field of view and where theobject is within the field of view, and the camera communicating aplurality of images to the processor; and the processor receiving theplurality of images and determining the object location and thesubsequent object location using the plurality of images.
 12. The systemof claim 11 further comprising the processor determining the objectvelocity using the plurality of images.
 13. The system of claim 10further comprising the object communicating the object location and thesubsequent object location to the processor.
 14. The system of claim 10further comprising: one or more inclinometers having a first measurementaxis parallel to the axis x_(p) and a second measurement axis parallelto the axis y_(p), and the one or more inclinometers communicating afirst inclination and a second inclination to the processor, where thefirst inclination describes an angle between the axis x_(P) and thelocal gravity vector and the second inclination describes an anglebetween the axis y_(P) and the local gravity vector; and the processorutilizing the first inclination and the second inclination to determinea current planar surface orientation.
 15. The system of claim 10 furthercomprising: one or more distance meters communicating one or moredistances to the processor where the one or more distances are parallelto the local gravity vector; and the processor utilizing the one or moredistances to determine a current planar surface orientation.